# Thermophysical Characteristics of the Ferrofluid in a Vertical Rectangle

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Method

_{1}) wall was basically maintained at 340 K and the temperatures of the left (T

_{2}), right (T

_{4}) and top (T

_{3}) walls were maintained at 300 K. The temperatures of the right (T

_{4}) and left (T

_{2}) walls were varied to 290 K, 300 K and 310 K. Magnetic field intensities ranged from 0 to 3000 A/m with an interval of 1000 A/m and were applied uniformly along the y-direction at the bottom wall. Velocities at the initial conditions were assumed to be 0 in both the x- and y-directions, and all walls were under no-slip boundary conditions. Table 1 shows the numerical conditions.

^{5}as suggested by Hollands et al. [14]. The thermodynamic properties of the ferrofluids used in this study were based on the data supplied by the manufacturing company [15]. Table 2 lists the thermodynamic properties of the ferrofluids used in this study.

^{−5}K during 0.5 s. The elapsed time was t = 20,000 s. The conservation equations for mass, momentum and energy were used to analyze the heat and flow characteristics of the ferrofluid in the vertical rectangle. In addition, the magnetization and Maxwell equations are used to predict the ferrofluid behaviors with consideration of the rotating effect of the nano-sized magnetic particles [16]. In addition, the governing equations were solved with several physical assumptions as well as the well-known Boussinesq approximation [17]. The local Nusselt number on the near heated wall (T

_{1}) was defined as an Equation (7) and the mean Nusselt number was calculated as an Equation (8), based on the research results of Seo and Lee [18]:

## 3. Results and Discussion

#### 3.1. Thermal-Flow Characteristics

_{1}= 340 K and T

_{3}= 300 K.

_{1}= 340 K and T

_{3}= 300 K. The thermal diffusion of the ferrofluid in the vertical rectangle was enhanced with the rise of the magnetic field intensities. This is because the magnetic volume force for the y-direction increased with the rise of the magnetic field intensities at the bottom wall of the rectangle [19].

#### 3.2. Nusselt Numbers and Heat Transfer Characteristics

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Comparisons of the isotherms with various magnetite volume fractions of ferrofluids at H = 0 A/m. (

**a**) 20,000 s; (

**b**) 4000 s.

**Figure 3.**Isotherms of ferrofluid with various intensities of magnetic field at the magnetite volume fractions of 1.5%, 3.0% and 4.5%. (

**a**) Volume fraction (1.5%) at 20,000 s; (

**b**) Volume fraction (3.0%) at 20,000 s; (

**c**) Volume fraction (4.5%) at 20,000 s; (

**d**) Volume fraction (3.0%) at 4000 s.

**Figure 4.**Mean Nusselt numbers of the ferrofluid in the vertical rectangle with the magnetite volume fractions.

**Figure 5.**Both local and mean Nusselt numbers and isotherms of ferrofluid in the vertical rectangle with various temperatures of the left wall at H = 0 A/m and magnetite volume fraction of 3.0%. (

**a**) Local Nu at bottom wall; (

**b**) Local Nu at top wall; (

**c**) Mean Nu at top and bottom walls; (

**d**) Isotherms.

Properties | Conditions |
---|---|

Magnetic field intensities (A/m) | 0, 1000, 2000, 3000 (Normal) |

Temperature of the left wall, T_{2}, T_{4} (K) | 270, 280, 290, 300 (Normal), 310, 320 |

Temperature of the left wall, T_{1} (K) | 340 (Normal) |

Temperature of the left wall, T_{3} (K) | 300 (Normal) |

Properties | Water | FF1 | FF2 | FF3 |
---|---|---|---|---|

Volume fraction of magnetite (%) | 0 | 1.5 | 3.0 | 4.5 |

Nominal particle diameter (nm) | – | 10 | 10 | 10 |

Density at 298 K (kg/m^{3}) | 946.9 | 1044 | 1142 | 1240 |

Viscosity at 300 K (mPa·s) | 0.854 | 1.9 | 2.95 | 4.0 |

Saturation magnetization (mT) | – | 9.2 | 18.3 | 27.5 |

Prandtl number | 6.1 | 13.2 | 20.3 | 27.4 |

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**MDPI and ACS Style**

Seo, J.-H.; You, B.-H.; Kwen, S.-S.; Lee, D.-Y.; Lee, M.-Y.
Thermophysical Characteristics of the Ferrofluid in a Vertical Rectangle. *Entropy* **2015**, *17*, 903-913.
https://doi.org/10.3390/e17020903

**AMA Style**

Seo J-H, You B-H, Kwen S-S, Lee D-Y, Lee M-Y.
Thermophysical Characteristics of the Ferrofluid in a Vertical Rectangle. *Entropy*. 2015; 17(2):903-913.
https://doi.org/10.3390/e17020903

**Chicago/Turabian Style**

Seo, Jae-Hyeong, Byoung-Hee You, Sang-Seuk Kwen, Dong-Yeon Lee, and Moo-Yeon Lee.
2015. "Thermophysical Characteristics of the Ferrofluid in a Vertical Rectangle" *Entropy* 17, no. 2: 903-913.
https://doi.org/10.3390/e17020903